If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^{2}}\right)^{10}$ is $405,$ then $|k|$ equals 

If $a^3 + b^6 = 2$, then the maximum value of the term independent of $x$ in the expansion of  $(ax^{\frac{1}{3}}+bx^{\frac{-1}{6}})^9$ is, where $(a > 0, b > 0)$

In the binomial expansion of ${\left( {a - b} \right)^n},n \ge 5,\;$ the sum of $5^{th}$ and $6^{th}$ terms is zero , then $a/b$ equals.

Let for the $9^{\text {th }}$ term in the binomial expansion of $(3+6 x)^{n}$, in the increasing powers of $6 x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is $n_{0}$. If $k$ is the ratio of the coefficient of $x ^{6}$ to the coefficient of $x ^{3}$, then $k + n _{0}$ is equal to.

Given that $4^{th}$ term in the expansion of ${\left( {2 + \frac{3}{8}x} \right)^{10}}$ has the maximum numerical value, the range of value of $x$ for which this will be true is given by

The middle term in the expansion of ${\left( {3x - \frac{{{x^3}}}{6}} \right)^9}$ are :-